Solve the equation 2 tan x 2 + 3 = 0 for the domain 0 ∘ ⩽ x ⩽ 540 ∘ .

Given that, the trigonometric equation is 2 tan x 2 + 3 = 0 for 0 ° ⩽ x ⩽ 540 ° Let A = x 2 2 tan A + 3 = 0 tan A = - 3 2 A = tan - 1 - 3 2 = - 40 . 89 ° Looking at the sketch and using the symmetry of the curve: the first value is 180 ° - 40 . 89 ° = 139 . 11 ° The second value is 360 ° - 40 . 89 ° = 319 . 11 ° Using A = X 2 x = 139 . 11 ° × 2 x = 278 . 22 ° [Note: x = 319 . 11 × 2 i . e . x = 638 . 22 ° is out of range 0 ° ⩽ x ⩽ 540 ° ] x = 278 . 2 ∘ (correct to 1 decimal place).

Solve the equation 2 sin x 3 + π 4 = 1 for the domain 0 < x < 4 π .

Given that, 2 · sin x 3 + π 4 = 1 , 0 < x < 4 π ⇒ sin x 3 + π 4 = 1 2 ⇒ sin x 3 + π 4 = sin π 4 ⇒ x 3 + π 4 = π 4 ⇒ x 3 = π 4 - π 4 ∴ x = 0 Now, 2 sin x 3 + π 4 = 1 ⇒ sin x 3 + π 4 = 1 2 ⇒ sin x 3 + π 4 = sin 3 π 4 ⇒ x 3 + π 4 = 3 π 4 ⇒ x 3 = 3 π 4 - π 4 ⇒ x 3 = 2 π 4 = π 2 ⇒ x 3 = π 2 ∴ x = 3 π 2

Solve the equation 2 sin x = cos x for 0 ∘ ⩽ x ⩽ 360 ∘ .

Given that, 2 sin x = cos x ⇒ sin x cos x = 1 2 ⇒ tan x = 1 2 ∴ Now, x = tan - 1 1 2 ∴ x = tan - 1 ( 0 . 5 ) ∴ x = 26 . 565 ° or 180 ° + 26 . 565 ° ∴ x = 26 . 565 ° or 216 . 565 ° Where 0 ≤ x ≤ 360 °

Solve the equation 2 sin x - 3 cos x = 0 for 0 ∘ ⩽ x ⩽ 360 ∘ .

Given that, 2 sin x - 3 cos x = 0 , 0 ≤ x ≤ 360 ° ∴ 2 sin x = 3 cos x ∴ sin x cos x = 3 2 ∴ tan x = 3 2 ⇒ x = tan - 1 3 2 ⇒ x = 56 . 3 ° or 236 . 3 °

Solve the equation 4 sin x + 7 cos x = 0 for 0 ∘ ⩽ x ⩽ 360 ∘ .

Given that, 4 sin x + 7 cos x = 0 , 0 ≤ x ≤ 360 ° ⇒ 4 sin x = - 7 cos x ⇒ sin x cos x = - 7 4 ∴ tan x = 7 4 ∴ x = tan - 1 - 7 4 ∴ x = 119 . 64 ° or 119 . 64 + 180 ° ⇒ x = 119 . 64 ° or x =299.64°

Solve the equation 3 cos 2 x - 4 sin 2 x = 0 for 0 ∘ ⩽ x ⩽ 360 ∘ .

3 cos 2 x - 4 sin 2 x = 0 for 0 ° ⩽ x ⩽ 360 ° Divide both sides by cos 2 x and use tan θ = sin θ cos θ 3 - 4 tan 2 x = 0 4 tan 2 x = 3 Let A = 2 x tan A = 0 . 75 tan - 1 0 . 75 = A A = 36 . 86 . . . . . . ∘ Using the symmetry of the curve, the second value is A = 180 ° + 36 . 86 … ° = 216 . 86 … ° the third value is A = 360 ° + 36 . 86 … ° = 396 . 86 . . . . . ∘ the fourth value is A = 540 ° + 36 . 86 … ° = 576 . 86 . . . . . . ∘ As A = 2 x : 2 x = 36 . 86 . . . . ° x = 18 . 4 ° 2 x = 216 . 86 . . . . ° x = 108 . 4 ° 2 x = 396 . 86 . . . . ° x = 198 . 4 ° 2 x = 576 . 86 … ° x = 288 . 4 ° Solutions are: 18 . 4 ° , 108 . 4 ° , 198 . 4 ° , 288 . 4 ° to 1 decimal place.

Solve the equation sin x cos x - 60 = 0 for 0 ∘ ⩽ x ⩽ 360 ∘ .

Given that, sin x · cos ( x - 60 ) = 0 ⇒ sin x · cos ( x - 60 ) = 0 ⇒ sin x = 0 or cos ( x - 60 ) = 0 ⇒ sin x = 0 x = n π , where n ∈ z ∴ x = π , 2 π and cos ( x - 60 ) = 0 ∴ x = n π 2 Where n is odd multiple of π 2 Here, x - 60 ° = 90 ° ⇒ x = 90 ° + 60 ° ⇒ x = 150 ° and x - 60 = 270 ° ∴ x = 270 + 60 ° ∴ x = 330 °